$ 



LIBRARY OF CONGRESS, t 
# - # 

$ -0*4 ,S45 i . J 

<&«fc,.yAe^ V,.. J 



j UNITED STATES OF AMERICA.) 



Life Agent's Aid, 



FOR .THE USE OF 



THE xVGEXTS 



UXION MUTUAL 



LIFE INSURANCE COMPANY. 



DIRECTORS OFFICE : 

27 COURT ST.. BOSTON. MASS. 



By hexey w: smith. 



It t fo Work: 



r a x c e m o x J 

] 'in . 



Life Agent's Aid, 



FOR THE USE OF 



THE AGENTS 



UNION MUTUAL 



LIFE INSURANCE COMPANY. 



directors' office: 
27 COURT ST., BOSTON, MASS. 



By HENRY W. SMITH. 



8«fa iorfe: 

PUBLISHED BY THE INSURANCE MONITOR. 
1869. 






Entered according to Act of Congress in the year 1869, by C. C. Hine, 

in the Clerk's Office of the District Court for the 

Southern District of New York. 



THE LIFE AGENT'S AID. 



It will be the province of this essay to ascertain 
the sources from which the surplus of Life Assur- 
ance Societies arises; to state the usual methods that 
have been adopted for its division; and to give, 
illustrated by arithmetical processes, an intelligible 
explanation of the Contribution Plan recently adopt- 
ed by a considerable number of the American Com- 
panies. In furtherance of this design we deem it 
essential to call the attention of the reader to the 
fundamental principles upon which Mutual Life 
Assurance Societies are founded. 

The science of Life Contingencies is based upon 
the doctrine of chances. These, paradoxical as it 
may seem 3 are subject to a law, the operation of 
which is as regular as that of gravitation. If we 
examine the history of any large number of happen- 
ings, we shall find that they have recurred with sur- 
prising uniformity. ; For example, the number of 
aces which come up in throwing a die ten thousand 
times, indicates with a good degree of accuracy 
how many will come up in the next ten thousand 
throws. Extended data, in relation to the number 
of casualties of any description that have taken 
place in a given locality, will enable us to predict 
with tolerable certainty in respect to the recurrence 
of similar events in the future. This is well illus- 
trated by the number of violent deaths that occur- 



red in Great Britain during five consecutive years. 
They were as follows : — 

In 1843 1849 1850 1851 1852 

13,551 13,324 13,987 13,559 14,475 

Of this number there died from poison : — 

In 1848 1849 1850 1851 1852 

569 526 553 528 553 

Murders seem to be subject to the same law of 
average. We give statistics from M. Quetelot, 
relating to the number of murderers in France 
brought to justice during the following years : — 

In 1826 1827 1828 1829 1830 1831 

241 234 227 231 207 266 

While, at the end of the year 1852, it would 
have been easy to predict the number of deaths 
by casualty that would have occurred in Great Brit- 
ain in 1853 ; or in 1831 to foretell the number of 
murderers that would be brought to justice in France 
in 1832 ; it would have been impossible to establish 
the identity of the parties who were to be in any way 
interested in these transactions. 

It would be equally difficult to predict the dura- 
tion of a single human life. Nevertheless, but few 
things are more determinate than the average du- 
ration of life in a multitude of individuals. One 
might well be amazed at the consonance of tabulated 
results, if he considers for a moment that the obser- 
vations from which they were deduced were made 
at widely different times, and in countries which 
seemed to be under entirely different sanitary con- 
ditions. 

The average duration of life in Rome, thirteen 



hundred years ago, was the same as in Great Brit- 
ain at the present time. Among the English nobles, 
" the expectation of life" at the age of eighty-four 
is about four years, and that of the fisherman at 
Ostend is precisely the same. With these results, the 
records of M. Deparcieux, made over a century ago, 
substantially agree. " The expectation," as deduced 
by Halley, from observations made in the seven- 
teenth century, is but a little less, and is concordant 
with the " Combined Experience of Seventeen Com- 
panies," tabulated by Jenkin Jones in 1843. The 
Carlisle Table, computed by Mr. Milne from obser- 
vations made in an English town by Dr. Heysham 
during the eight years prior to 1787, and the new 
American Table published in the Report of the Mas- 
sachusetts Insurance Commissioner for 1868, are 
strikingly coincident. 

The idea of ministering to the wants of mankind 
by making extended observations of births and 
deaths, and applying the theory of probabilities 
thereto, is of ancient origin. It did not in earlier 
times seem to meet with general appreciation. 
The older societies, ignoring very trustworthy data, 
evidently guessed at their rates of premium. They 
set sail like a ship without compass or chart, and 
in some instances were lucky enough to steer clear 
of the hidden reefs and to prosper on their voyage. 
The Amicable Society, which was organized in 1706, 
charged the same rates of premium on all ages be- 
tween twelve and forty-five. This was said to be 
an entrance fee of £3 15s. per cent., and an annual 
premium of £5 per cent. Mr. Babbage endeavored, 
by examining the earlier records of this Society, 
to ascertain why £5 per cent, was taken, and con- 
cluded that it arose from the fact, that in the city 
of London the deaths were about one in twenty. 
At the present time it seems very strange that this 



8 

Society should have been content to grope in 
the dark, when the Breslau Table, published thir- 
teen years previous to its organization, defined with 
a good degree of accuracy the probability of surviv- 
ing at any age. " When," says Dr. Gouraud in 
"The History of the Calculus of Probabilities," "in 
1693, an English mathematican of the highest order, 
proceeding in turn to study, in the obituary returns 
of London and Breslau, the general laws of human 
mortality, published on this subject a Memoir, 
which is even now read with admiration, absolutely 
no one took heed of it. * * * Useless 

instructions ! Buried in the vast and rich collection 
of Memoirs of the Royal Society of London, the ad- 
mirable labors of Halley were only to be discovered 
by posterity." 

Something like half a century after the organiza- 
tion of the Amicable Society, Thomas Simpson, a 
self-taught mathematician, after having established 
a reputation by the publication of various works on 
annuities, delivered a course of lectures in London, 
in which he excited the attention of the public by 
announcing the possibility of constructing a table 
of Assurance premiums, accurately graduated to the 
mortality of each age. Impelled by these state- 
ments, James Dodson turned his attention to the 
subject, and actually computed a table of rates. This 
circumstance, in combination with the increased dif- 
fusion of information on the subject of life contin- 
gencies, led, in 1762, to the establishment of the 
Equitable Society. This was the inauguration of a 
new era in the history of Assurance. Nothing that 
had previously existed was either safe or practicable. 
The new system, founded upon data obtained from 
actual observation, was established upon a scientific 
basis, and possessed the elements of stability. 

This pencil of light stimulated further research. 



The immense practical value of the subject was ac- 
knowledged, and to its consideration the attention of 
some of the ablest men in the kingdom was turned. 
Arrangements were made for more extended ob- 
servation. The government aided in procuring the 
necessary statistical information. New tables were 
computed, each one verifying the financial safety 
of the results already attained. Extended compari- 
sons were made between data obtained from various 
sources. Those tables found to be best graduated to 
the actual mortuary experience were accepted as 
standards, while others gradually grew obsolete. 
This process of comparing and eliminating has ren- 
dered the Rates of Mortality which are now in 
general use very trustworthy representatives of the 
decrement of human life. 

A Life Assurance Society is founded on two as- 
sumptions, viz. : that the mortality among its mem- 
bers will be the same as that of the " Rate" which 
it has adopted as its basis ; and that a certain per- 
centum of interest can be realized on its invest- 
ments. It is not a difficult, although a somewhat 
intricate, matter to compute upon these conditions 
a table of premiums which will make the business 
of Assurance practicable. A Company thus organ- 
ized, if the original assumptions are just realized, is 
sure to be exactly solvent. It can pay every loss 
as it matures, and in paying the last will use every 
remaining dollar of its fund. If more favorable 
conditions are experienced, a surplus is realized; 
if less favorable, its bankruptcy is only a question 
of time. In order to accomplish its beneficent mis- 
sion, the security of a Life Assurance Company 
should be as absolute as human forethought can 
make it. Hence, as a premium after a contract is 
made cannot be altered, the original assumptions 
should be such as will enable it to charge a larger 



10 



rate than will be required daring the lifetime of 
the youngest member, What is not needed can be 
returned periodically to the assured. Practically, the 
selection of a rate of mortality, provided either of 
the standard tables are taken, is of less importance 
than the assumption of a per-centum of interest. 
The difference of one single unit in this element 
might, in time, either insure the prosperity of a Com- 
pany or drive it into bankruptcy. By almost uni- 
versal consent the rate adopted by the American 
Companies is four per cent. ; while the English Com- 
panies generally assume two and one-half or three 
per cent. To guard against adverse contingencies, 
and to provide for the expenses incident upon the 
conduct of business, it is customary to add to the net 
premium a margin or loading of from twenty to forty 
per cent. This constitutes what is usually known 
as the office premium. 

In order to show the relation of the premium to 
the table of mortality, we will ascertain the net 
annual amount which would be paid to secure an 
assurance of one dollar, at the age of ninety years, 
assuming four per cent, interest, by the most intelli- 
gible arithmetical process by which the computation 
can be made. 

By referring to the "Actuaries' Rate of Mortality," 
we find the number living and dying at the age of 
ninety, and at each subsequent year, is as follows : — 



Age. 


No. Living. 


No. Dying.. 


Age. 


No. Living. 


No. Dying. 


90 


1319 


427 


95 


89 


52 


91 


892 


322 


96 


37 


24 


92 


570 


231 


97 


13 


9 


93 


339 


155 


98 


4 


3 


94 


184 


95: 


99 


1 


1 



11 

The first step is to ascertain the present value of 
an annuity of one dollar at the age of ninety years, 
the first payment of which is to be made at once. 
This is equal to the payment for the first year, plus 
the present value of all future payments. 

If thirteen hundred and nineteen persons were 
each to receive a payment of one dollar at the com- 
mencement of every year during the remainder of 
life, the first payment being made at once, is evi- 
dently equal to $1319. At the commencement of 
the ninety-first year there would be but eight hun- 
dred and ninety-two of the thirteen hundred and 
nineteen persons alive ; hence, but $892 would be 
paid. If, at the commencement of this year, the 
Company were to make an adjustment for both 
years, it would pay, in addition to the $1319 then 
due, a sum which, improved at the assumed rate 
of interest, would amount at the end of one year 
to $892 ; or, in other words, its present worth. As 
every one of the annuitants has an equal claim upon 
this sum, it must be divided into thirteen hundred 
and nineteen parts. The share of each would be 
the present worth of y 8 ^ 9 -^ of one dollar, or ££fa of 
unity, multiplied into the present value, one dollar, 
payable in one year. At the beginning of the third 
year there would be but five hundred and seventy 
of the thirteen hundred and nineteen living ; conse- 
quently, but $570 would be paid. Commuting this 
payment at the commencement of the ninetieth year, 
the share of each one would be • £££§ of unity, multi- 
plied into the present value of one dollar, payable 
two years hence. The present worth of all pay- 
ments ascertained in this manner constitutes the 
value of a life annuity. 

This process is completed in the following table : — 



12 



TABLE I. 







Age, . . . Ninety. 








The value 






Age. 


Number 


of each one's 


Present worth of One 


Present worth of 


Living. 


share, unim- 
proved. 


Dollar. 


each Payment. 


90 


1319 


i|ia x $1.00000000 = 


: $1.0000000 


91 


892 


-fWs x .96153846 = 


.6502536 


92 


570 


Trr-9 x .92455625 = 


.3995429 


93 


339 


-i¥r 9 - x .88899636 = 


.2284885 


94 


184 


-i¥rg x .85480419 = 


.1192448 


95 


89 


rffs x .82192711 = 


.0554598 


96 


37 


7 fig x .79031353 = 


: .0221696 


97 


13 


tH-9 x .75991781 = 


.0074897 


98 


4 


T- 3 -f9 x .73069020 = 


.0022168 


99 


1 


rwt9 x .70258674 = 


.0005326 


Presei 


it valu< 


3 of an annuity of One Dollar, $2.485398 3 



The next step will be to ascertain the amount of 
the net single premium on a whole life policy of 
one dollar. 

According to the Rate of Mortality, four hundred 
and twenty-seven, out of thirteen hundred and nine- 
teen persons living at the age of ninety, will die 
within one year. If each one were assured for one 
dollar, and the losses adjusted at the end of the year, 
$427 would then be due, and the share of each assurant 
entered at the commencement of the year would 
be i^th part of the $427 or T 4 ^ of one dollar. 
If the payment of the premium is made at the com- 
mencement of the year, such a sum must be taken as 
will, improved at the assumed rate of interest be equal 
to $427 at the end of the year, or, in other words, its 
present value ; and the share of each assurant will 
be tW-9 of unity, multiplied by the present value of 
one dollar, payable one year hence. During the 
second year three hundred and twenty-two will die. 
If the whole number living at the commencement 
of the ninetieth year were assured for two years. 



13 



paying the assessment of both years at the outset, 
the whole amount due would be the present value 
of $427, payable in one year, plus the present value 
of $322, payable in two years. The share of each 
assurant would be -ffifa of unity, multiplied into the 
present value of one dollar, payable in one year, 
added to T 3 3 2 T 2 T of unity, multiplied into the value of 
one dollar, payable in two years. Adjusting in this 
manner, at the commencement of the ninetieth year, 
all losses for the remainder of life, the share of each 
one will be the net single premium. This is always 
equal to the present value of each individual's share 
of the amount paid for losses at the end of the first 
and of every subsequent year. 
We have completed this process in the table below: 

TABLE II. 







Age, 


. . . Ninety. 








Assessment 








Number 


of each Assur- 


Present value of One 


Present value of each 


Age. 


Dying. 


ant at the end 
of the year. 


Dollar. 


Year's Payment. 


90 


427 


ffis » 


$.96153846 = 


r $.3112788 


91 


322 


■ffls > 


: .92455621 = 


= .2257067 


92 


231 


•fit* > 


c .88899636 = 


r .1556922 


93 


155 


-M% > 


c .85480419 = 


= .1004507 


94 


95 


iff* > 


c .82192711 = 


= .0591868 


95 


52 


iih > 


c .79031453 = 


= .0311572 


96 


24 


_24_ . 
13 19 > 


c .75991781 = 


= .0138272 


97 


9 


13 19 > 


< .73069020 = 


= .0049857 


98 


3 


•nfrg > 


c .70258674 = 


= .0015980 


99 


1 


t^rs > 


( .67556417 = 


= .0005122 


Ne 


t single 


5 premium 


for one dollar, 


$.9043955 



The present value of a life annuity of one dollar 
is equal to the present worth of one dollar per 
annum, to be received at the commencement of 
each year during life. The net annual premium is 
such an amount paid at the commencement of each 



14 

year, that the present value of the first and all subse- 
quent premiums shall be equal to the net single pre- 
mium. As the value of a life annuity is the present 
worth of a certain sum to be received at the com- 
mencement of each year, so the net single premium 
is the present worth of all the net annual premiums. 
Hence, the present value of an annuity of one dollar 
holds the same relation to one dollar, that the net 
single premium for one dollar holds to the net an- 
nual premium. Taking the value of an annuity of 
one dollar, and the net single premium for one dol- 
lar already obtained, and we have the following 
proportion : — • 

As the present value of the annuity is to one 
dollar, so is the net single premium to the net 
annual premium ; or, 

$2.4853983 : $1 : : .9043955 : .36388312 

* The fourth term is the net annual premium 
for whole life assurance. It is, owing to loss of 
decimals, a little less than it should have been. The 
premium, more exactly, at this age is $.36388844; 
consequently, the premium for one thousand dollars 
will be $363.88844. 

The correctness of this process is susceptible of 
proof. If we take at the end of each year from the 
fund made up from the premiums for the year, and 
the balance from preceding years, improved at four 
per cent., the amount of losses paid, we can meet 
every claim as it matures, and at the end of the 
ninety-ninth year, upon the payment of the last loss, 
the fund will be exhausted. 

* The method which we have employed in computing this premium lies at the 
foundation of all calculations involving premiums and annuities. There are 
extended abridgments by which the labor of computation is materially light- 
ened, but our limits do not permit us to introduce them. We would refer the 
reader, for further information on the subject, to any standard work on annui- 
ties. " The Agent's Monetary and Life and Valuation Tables," by D. P. Fackler, 
contains in a very compact form the formulas and tables in general use. 



15 

The following is a complete statement of the 
yearly account between the Company and thirteen 
hundred and nineteen members, assured for one 
thousand dollars each, by net annual premiums, 
entered at the age of ninety, upon the supposition 
that the assumptions upon which the premiums are 
based are just realized : — 

FIRST TEAR. 

Living, 1319; dying, 427. 

1319 premiums are $479,968,852 

Improved at 4 per cent, interest, is 499,167.607 

Deduct losses for the year 427,000.000 

And the balance remaining is 72,167.607 

SECOND YEAR. 

Living, 892 ; dying, 322. 

The balance from the end of last year $ 72,167.607 

And 892 premiums ,. 324,588.488 

Improved at interest, is 412,626.339 

Deduct losses of 322,000.000 

And balance remaining is 90,626.339 

THIRD YEAR. 

Living, 570 ; dying, 231. 

The balance from the end of last year $ 90,626.339 

. And 570 premiums 207,416.411 

Improved at interest, is 309,964.459 

Deduct losses of , 231,000.000 

And the balance remaining is 78,964,459 

FOURTH YEAR. 

Living, 339 ; dying, 155. 

The balance from the end of last year $ 78,964.459 

And 339 premiums 123,358.181 

Improved at interest, is 210,415.546 

Deduct losses of. 155,000.000 

And the balance remaining is 55,415.546 



16 



FIFTH YEAR. 

Living, 184; dying, 95. 

The balance from the end of last year $ 55,415.546 

And 184 premiums 66,955.473 

Improved at interest, is 127,265.859 

Deduct losses of 95,000.000 

And the balance remaining is 32,265.859 

SIXTH YEAR. 

Living, 89 ; dying, 52. 

The balance from the end of last year $ 32,265.859 

And 89 premiums 32,386.071 

Improved at interest, is 67,238.008 

Deduct losses of 52,000.000 

And the balance remaining is 15,238.008 

SEVENTH YEAR. 

Living, 37 ; dying, 24. 

The balance from the end of last year $ 15,238-008 

And 37 premiums 13,463.872 

Improved at interest, is 29,849.955 

Deduct losses of 24,000.000 

And the balance remaining is 5,849.955 

EIGHTH YEAR. 

Living, 13 ; dying, 9. 

The balance from the end of last year ,.. $ 5,849.955 

And 13 premiums 4,730.550 

Improved at interest, is 11,003.725 

Deduct losses of 9,000.000 

And balance remaining is 2,000,725 

NINTH YEAR. 

Living, 4 ; dying, 3. 

The balance from the end of last year $ 2,000.725 

And 4 premiums 1,455.534 

Improved at interest, is 3,597.650 

Deduct losses of 3,000.000 

And balance remaining is 597.650 



17 



TENTH YEAR. 

Living y 1 ; dying, 1. 

The balance from the end of last year $ 597.650 

And 1 premium 363.888 

Improved at interest, is 1000.000 

Deduct loss of. 1000.000 

And balance remaining is 000.000 

The reader cannot fail to have observed that in 
the statement just given, the original assumptions 
in respect of mortality and interest are just realized, 
and that the Company is able to fulfill all its con- 
tracts, and no more. There is remaining at the end 
of every year, save the last, a considerable portion of 
the original premium unexpended, after providing 
for all losses. This constitutes what is usually called 
the reserve, or the cost of re-assurance. The portion 
of this fund belonging to an individual policy aL .ue 
close of any year, is indicated by dividing the balance 
on hand, after paying all losses, by the number of as- 
surants alive at the commencement of the next year. 
It would be well to notice the fact that in this ex- 
ample, while after the second year the whole amount 
of the reserve decreases, that of each policy, owing 
to the withdrawals by death, increases. At the end 
of the first year the reserve on a single policy was 
$80.90 ; at the end of the fifth year it had increased 
to §362.54, and to $597.60 at the end of the ninth. 

To illustrate more clearly the origin of the sur- 
plus, we would consider still further the assumptions 
upon which Companies are founded. 

I. Taking a case in which — 

1st. The mortuary experiences of the Company 
exceeds the decrement of the assumed 
rate of mortality ; 

2d. The rate of interest realized is less than 
that assumed; 



18 

3d. The margin or loading is insufficient to pro- 
vide for current expenses. 

If the conclusions already reached are correct, it 
will need no argument to show the inherent un- 
soundness of a Company which is in the condition 
just described. The failure to realize the assumed 
rate of interest would alone leave the reserve inad- 
equate, and the state of the Company would be ren- 
dered still worse by the encroachments upon the 
safety fund to meet losses and to provide for ex- 
penses. • If it paid the earlier claims, it would inevi- 
tably fail to meet the later ones. By the accession 
of new members it might for years be able to meet 
its liabilities, but its ultimate failure would be as 
certain as the coming of seed-time and harvest. 

II. Let us consider a case in which — 

1st. The mortuary experience is less than the de- 
crement of the assumed rate of mortality; 
2d. The rate of interest actually realized is 

greater than that assumed ; 
3d. The margin or loading is more than suffi- 
cient to provide for current expenses. 

Then a surplus would remain after providing the 
necessary reserves and meeting all claims for losses. 
We are happy to be able to state that this is the 
case with a large majority of the American Compa- 
nies. By very safe assumptions, prudent manage- 
ment, judicious investments, and careful medical se- 
lection of risks, their balance-sheets usually show, 
upon the periodic examination of their affairs, a con- 
siderable amount of surplus. The troublesome 
problem is, 

HOW OUGHT THIS SURPLUS TO BE DlVIDED ? 

' Various systems have been devised in answer to 
this interrogatory. Nearly every office has a plan 



19 

which, while it may bear a general resemblance to 
that of some other,, is so modified as to be charac- 
teristic ; and every Company, with not a little perti- 
nacity, maintains that its system is fully as satisfac- 
tory as that of any of its neighbors. In accordance 
with their leading features, these plans may be 
divided as follows : — 

1st. By a per-centage on the sums assured ; 
2d. By a per-centage on the sums assured, and 

on all reversionary additions thereto ; 
3d. By a per-centage on all premiums paid, and 

an additional per-centage on all previous 

dividends ; 
4th. By a per-centage upon all premiums paid ; 
5th. By reversionary additions to the policy; 
6th. By the Contribution Plan. 

The first and second of these systems are peculiar 
to the English Companies, and the sixth to Ameri- 
can Companies. 

The query would naturally be suggested as to 
which of these systems is the most desirable. The 
answer would be, that which establishes the most 
equitable relations between the Company and the 
individual policyholder. 

Mr. Sang, an English actuary, while the business 
of Life Assurance in this country was in its infancy, 
very justly remarked : — 

" Could the mortality amongst the members be 
accurately predicted, and the profits of investments 
foretold, the premiums could be computed so as just 
to meet the engagements ; in which case there would 
never be any surplus fund ; each member would 
receive exactly the benefit to which he is entitled ; 
and, be it observed, no member can receive more 
than his share of the benefit, if not at the expense 



20 

of some other member who receives less. For secu- 
rity, somewhat more than the riet value is charged 
as premium for assurance, and the excess of the 
actual over the net premium goes to form the profit 
fund. This excess, in fact, constitutes the profit on 
a policy." 

These views indicate that what are usually termed 
profits, are simply the over-payments of the policj 7 - 
holders. The fund arises, as we have already seen, 
partly from the loading, partly from the excess of 
interest, and partly from a more favorable mortuary 
experience than that assumed. In equity this fund 
must be classed as a liability in which each policy- 
holder has an interest. It is customary among busi- 
ness men to keep an exact account of all liabilities, 
and to pass to the credit of each of their customers 
whatever balance may be his due. Should a corpo- 
ration endeavor to divide its profits by giving to 
each stockholder an equal amount, regardless of 
stock owned by each, it would soon find itself 
checked by a legal injunction, and, if necessary, its 
affairs would be placed in the hands of a receiver. 
A Mutual Life Assurance Company is a corporation 
in which the assurants are the stockholders, and 
their individual interest in the general fund varies 
with the size, age and class of their policies. From 
what source can a society of this kind derive any 
special rights or privileges in this matter not accord- 
ed to individ uals or other corporations ? Can it claim 
exemption from those usages which the common 
law has established between party and party ? As 
it is as much bound to preserve equitable relations 
between itself and its various members as it is to 
maintain its own solvency, can it do any more or 
any less than to return to each one of the assured 
whatever of the surplus he may have contributed ? 



21 

The system of division by which this is accom- 
plished is known as the Contribution Plan.* 

To gain a clear comprehension of this plan, it 
will be necessary to ascertain the sources from which 
the surplus fund has been derived : — 

1st. From Interest. — It is fair to presume that 
the funds paid by each individual policyholder 
have been improved in common with the funds 
paid by all the other policyholders ; consequently, 
^ach one will be entitled to receive the same per- 
centage of excess. 

2d. From the Loading. — Of all matters connected 
with an Assurance society requiring accuracy of 
adjustment, the determination of the proportion of 
profit arising from this source due each assurant is, 
perhaps, the most difficult. The expenses during 
successive years is subject to considerable variation. 
Companies differ so materially in the details of their 
methods of transacting business, that it is impossi- 
ble to give general directions. 

3d. From the Increased Yitality. — The ratio of 
mortality given by the table is the basis from which 
the premium is computed. If the mortality experi- 
enced at any age was greater than that assumed, it 
would be just to charge the excess to the margin or 
loading of the premium at that age ; if less, each 
policyholder should be credited with the proportion 
of the excess, and it should be returned to him at 
the periodic distribution. 

* In the Massachusetts Insurance Report for 186S, on page cxv., Mr. Sanford 
remarks: — "Tho Contribution Plan was first applied to the distribution of sur- 
plus by Mr. Sheppard Homans, the eminent actuary of the Mutual Life Insurance 
Company of New York, who has attributed to Mr. David Parks Fackler, his 
then assistant, now a consulting actuary of the same city, the suggestion of its 
idea, and has shared with him the credit of it3 discovery and development. Tbe 
system was first applied to the distribution of the quinquennial dividend of this 
Company in 1863, since which time it has been adopted by many leading Amer- 
ican Companies." 



22 

The most comprehensible method of adjustment 
between the Company and the assurants is an open 
debit and credit, in which the Company keeps the 
books. For the first, second and eleventh years the 
statement would be as follows : — 



FIRST YEAR. 

Policy No. . Cr. 

1st. By premium for the year. 

2d. By interest at the rate actually realized from the Com- 
pany's investments. 



1st. To cost of assurance of the year. 
2d. To his share of the necessary expense. 
3d. To the reserve held at the end of the year. 
4th. To dividend to balance. 



Dr. 



SECOND YEAR. 



1st. By reserve held at the end of the first year. 

2d. By premium for the year. 

3d. By interest on the sum of the premium and reserve. 



Cr. 



Br. 



Cr. 



1st. To cost of assurance for the year. 
2d. To expenses. 

3d. To reserve at the end of the year. 
4th. To dividend to balance. 

ELEVENTH YEAR. 

(On a Policy by Annual Payments.) 

1st. By reserve held at the end of the tenth year. 
2d. By premium for the year. 
3d. By interest. 

(On a Policy by Ten Annual Payments.) 

1st. By reserve held at the end of the tenth year. 
2d. By interest. 

Dr. 
1st. To cost of assurance for the year. 
2d. To expenses. 

3d. To reserve at the end of the year. 
4th. To dividend to balance. 



23 

r 

On the credit side of the account we gave the 
statement for a policy by annual payments, and for 
a policy by ten annual payments. This was done 
to show the items passed to the credit of paid-up pol- 
icies. The value of the elements used in this 
statement vary materially in different classes of bol- 
ides, but the form is applicable to all. 

It is evident that this subject is susceptible of an 
arithmetical elucidation. Taking, for example, a life 
policy for one thousand dollars, issued at the age 
of thirty-five, the payments on which are to be made 
by ten annual premiums, let us make the following 
assumptions, viz. :-^ 

1st. That the premium is based upon the " Com- 
bined Experience," or " Actuaries' Rate of Mortality, n 
and loaded thirty per cent. ; 

2d. That the expenses shall be equivalent to ten 
per cent, of the premium, taken at the commence- 
ment of the year ; 

3d. That the mortality shall be two-thirds of the 
decrement of the table ; 

4th. That the Company shall realize seven per 
cent, interest on its investments. ' 

Before proceeding further it will be necessary to 
ascertain the value of the various elements that en- 
ter into the account. 

1st. The Premium, — We have already computed 
and shown the nature of the net annual premium. 
The premium on a ten-payment whole-life policy 
furnishes financially an exact equivalent to the Com- 
pany for all the premiums on a whole-life policy by 
annual payments. It consists of two elements — the 
whole-life annual premium, and a sum which, im- 
proved at the rate of interest assumed in the com- 
putation of the tables, will purchase a deferred an- 
nuity equal to the whole-life annual premium, the 



24 



first payment of which is to be made at the com- 
mencement of the eleventh year. 

A deferred annuity is one in which the payments 
are to commence at some future specified time. It 
is usually purchased either by a single premium, or 
by annual premiums, the last of which is to be made 
at the commencement of the year previous to that 
upon which the first payment of the annuity is due. 
To illustrate the process by which the amount of 
these premiums is determined, the present values 
of the yearly payments of the annuity computed in 
Table I. are here introduced. 



Age. 


Present Value of One Dollar. 


Age. 


Present Value of One Dollar. 


90 
91 
92 
93 
94 


1.0000000 
.6502596 
.3995429 

.2284835 
.1192448 


95 
96 
97 

98 
99 


.0554598 
.0221695 
.0074897 
.0022158 
.0005396 



Present value of the first five payments $2.3975308 

Present value of the second five payments 0878674 

Present value of the ten payments 2.4853982 

The present value of a deferred annuity, entered 
at the age of ninety, the first payment of which is 
to be made at the commencement of the ninety- 
fifth year, is evidently the aggregate present values 
of one dollar to be received at the commencement of 
the ninety-fifth and of every subsequent year, or 
$.0878674, the present value of the second five pay- 
ments as indicated by the table. This is the net single 
premium. If paid by five annual premiums, the 
first of which is to be made at the beginning of the 
ninetieth year, such a sum must be taken annually 
as will, making due allowance for the chances of 
discontinuance by death, give a present value equal 



25 

to the single premium. In other words, the annual 
premium is equal to the yearly payment of a tem- 
porary annuity for five years, the value of which is 
$.0878674. The present value of a temporary an- 
nuity of one dollar, as indicated by the table, for 
the ninetieth and the four subsequent years, is 
$2.3975308. This amount holds the same relation 
to $1 that $.0878674, the single premium of a de- 
ferred annuity, does to the annual premium ; or 

$2.3975308 : $1 : : $.0878674 : $.03664449. 

The net annual premium on a whole-life policy 
for one thousand dollars, issued at the age of thirty- 
five, is $19.8665. The annual premium for a de- 
ferred annuity of $19.8665, the first payment of 
which is to be made at the beginning of the eleventh 
year, at the age of forty-five, computed by the 
method just given, is $22.1957. The sum ($19.8665 
4- $22.1957) of these two elements, or $42.0622, con- 
stitutes the premium on a whole-life policy by ten 
annual payments. A margin or loading of thirty 
per cent, gives $12.68 more, or an office premium of 
$54.68. This is about the rate usually charged by 
first-class companies. 

2d. The Expenses are, by our assumptions, to be 
equivalent to ten per cent, of the premium taken at 
the commencement of the year. This will_leave 
$49.21 as the effective office premium. 

For convenience in calculation we shall consider 
this amount as the office rate, and make no charge 
for expenses. This allowance is intended not only 
to defray the agency and office charges during the 
time in which the premiums are paid, but also to 
provide for the creation of a fund to meet the ex- 
penses of subsequent years. 

3d. The Reserve. — In the consideration of the 
statement of the account between the policyholders 



26 

and the Company made on page 17, it was remarked 
that the balance remaining at the end of each year, 
after the payment of all claims, was the reserve, and 
that the portion belonging to each policy could be 
readily ascertained by a simple division. This 
method of computation is not practicable. The net 
values of policies issued at different dates, and to 
persons of different ages, must be found separately. 
There are several methods employed for the valua- 
tion of life policies by annual payments. The fol- 
lowing is as convenient a plan as any : — 

The earlier assurant who has reached any given 
age must stand financially in the same relation to the 
Company as one who has been entered at that age, 
and the reserve retained must be such as will furnish 
an equivalent for the increase of the rate. The net 
single premium at the age of valuation is, as has 
been shown, equal to the value of an annuity of all the 
net annual premiums. The value of a policy at any 
time, together with the present value of all future 
premiums receivable, must be equal to the net single 
premium at the age of valuation. Hence the follow- 
ing rule : — 

From the net single premium at the age of valua- 
tion, subtract the present value at that age of a life an- 
nuity of the net annual premium at the age of entry : 

Net annual premium, age thirty-five $ 19.8665 

Net single premium, age thirty-six i 348.1712 

Value of an annuity of $1.00, age thirty-six 16.9475 

Value of an annuity of $19.8665, age thirty-six.... 236.6875 
Deduct from net single premium, and the balance 11.4837 

is the reserve at the expiration of the first year, just 
before the second premium is due. Another plan • 
of obtaining the same result is sometimes employed. 
It consists of finding the value of an annuity of the 
difference between the net premium at the age of 
entry, and that at the age of valuation : — 



27 

Net annual premium, age thirty-six $20.54404 

Net annual premium, age thirty-five 19.86653 

Difference .67751 

Value of an annuity of $.67751, age thirty-six... 11.4837 

The reserve on a ten-year policy is made up of 
two elements — the reserve on a simple life policy, 
and the sum of the accumulations of the payments 
of the deferred annuity. The accumulation of this 
payment the first year is the amount of the annuity, 
improved at the given rate of interest, divided by 
the probability of surviving* during the year : — 

Annual payment of deferred annuity of premium. $22.1957 

Improved at four per cent 23.0835 

Divided by |MIt> the probability of surviving.... 23.2999 

Add reserve of policy by annual payments 11.4832 

Reserve on a life policy by ten payments 34.7831 

To obtain the reserve at the end of the second year, 
we add to the reserve of a whole-life policy the ac- 
cumulations of both payments of the deferred annuity. 
At the end of the ten years, or at the age of forty- 
five, the reserve, the policy being paid up, will be 
just equal to the net single premium for whole-life as- 
surance at the age of forty-five. The reserves during 
the remainder of life are, in every instance, just equal 
to the net single premium at the age of valuation. 
This process can be verified as follows : — 
If the original assumptions are realized, the amount 
remaining is the reserve. The amount actually as- 
sured each year is not the amount of the policy, but 
in every instance the amount of the policy less the 
reserve held at the end of the year. On the policy 
for $1000 under consideration, the amount of the 
reserve at the end of the first year is $34.78, and the 
amount really at risk is not $1000, but $1000 less 
$34.78, or $965.22. If at the end of the year we take 
the cost of assurance from the net premium, im- 

* For further explanation of this matter, see reprint of Mass. Ins. Co. Reports, 
page 366. 



28 



proved at four per cent, 
cal with the reserve : — 



the balance will be identi- 



Amount assured $965,220 



Cost of assuring one dollar. 

Whole cost of assurance , 

Net premium 

Improved at four per cent — 
Less cost of assurance 



.0092877 
8.960 
42.062 
43.74 
34.78 



The difficulty encountered in obtaining the re- 
serve by this process is, that we cannot obtain the 
cost of assurance without the reserve. We can by 
considering $1000 the amount insured, and using 
balance as the reserve, by successive repetitions of 
the process, reduce the error to an infinitesimal. 

In Table III. the reserves on a ten-payment whole- 
life policy, issued at the age of thirty-five, are given. 
We also show the sum at risk each year by taking 
these reserves from the amount of the policy : — 







TABLE III. 








Age. 


Amount of 
Policy. 


Reserve at the end 
of the year. 


Amount at 
Risk. 


35 


$1,000 


less 


$ 34.78 


= 


$965.22 


36 


1,000 


a 


71.11 


= 


928.89 


37 


1,000 


tt 


109.07 


= 


890.93 


38 


1,000 


a 


148.74 


= 


857.26 


39 


1,000 


it 


190.23 


= 


809.27 


40 


1,000 


a 


233.64 


= 


766.36 


41 


1,000 


tt 


279.08 


= 


720.92 


42 


1,000 


ti 


326.66 


= 


673.34 


43 


1,000 


n 


376 45 


= 


623.55 


44 


1,000 


" 


428.57 


= 


571.43 


45 


1,000 


<< 


438.86 


= 


561 14 


46 


1,000 


tt 


449 35 


= 


550.65 


47 


1,000 


t< 


460.02 


= 


539 98 


48 


1,000 


it 


470.88 


= 


529.12 


49 


1,000 


tt 


481.91 


= 


518 09 


50 


1,000 


n 


493.10 


= 


506.90 



29 

The reserve which the laws of Massachusetts 
compel every Company doing business in that State 
to maintain, is based on the same assumptions that 
have been taken in computing Table III. The pol- 
icies are valued seriatim each year by the Com- 
missioner, and the aggregate amount of the reserves 
in each Company is compared with its net assets. 
Its solvency is thus tested every twelve months. 

4th. The Cost of Assurance. — By reference to the 
" Actuaries' Rate of Mortality," we find that the 
number living at the age of thirty-five is 85251, and 
that of this number 72 7 will die during the next 
year. The cost of assuring the monetary unit will be 
^ffly of that unit, or .0092877. This, multiplied by 
the number of units in any amount, will give at this 
age the cost of assuring that amount. By a similar 
method the cost of assuring any amount at any age 
can be readily ascertained. By our assumption the 
mortality was to be two-thirds of that of the table 
upon which the premium is based. The cost of 
assurance for the first year — the amount of the po- 
licy less the reserve, multiplied into two-thirds of 
cost of assuring one dollar — is given below : — 

Amount of the policy is $1000.00 

The reserve is , 34.78 

The amount at risk is.., 965.22 

The tabular cost of assuring one dollar is .0092877 

Two- thirds of this cost is 0061918 

The cost of assuring amount at risk is 5.9764 

In the table below this computation is given for 
the first sixteen vears. The amount at risk is taken 
from Table III. 



30 









TABLE 


IV. 










Age. 


Amount 
at Risk. 


Cost of Assur- 
ing Unity. 




Tabular Cost 
of Assurance. 


1 


Tw 


j-third 
lar 


> of Tabu- 
Cost. 


35 


$965.22 


> 


: .0092877 


= 


$8.96 


X 


■i 


= 


$5.98 


36 


928.89 


> 


: .0094849 


= 


8.80 


X 


o 
3 


= 


5.87 


37 


890.94 


x .0096867 


= 


8.62 


X 


2 
"3" 


= 


5.75 


38 


851.26 


x .0099064 


= 


8.43 


X 


2 
3 


= 


5.62 


39 


809.27 


x .0101311 


=3= 


8-18 


X 


3 


= 


5.46 


40 


766.36 


x .0103619 


= 


7.94 


X 


2 
3 


= 


5.29 


41 


720.92 


x .0106118 


= 


7.65 


X 


2 
3 


= 


5.10 


42 


673.34 


x .0108943 


= 


7.32 


X 


2 
3 


E=s 


4.89 


43 


623.55 


x .0112509 


= 


7.02 


X 


2 
3 


= 


4.67 


44 


571.43 


x .0116973 


= 


6.68 


X 


§ 


= 


4.46 


45 


561.14 


x .0122120 


= 


6.85 


X 


i 


= 


4.57 


46 


550.65 


x .0128389 


= 


7.07 


X 


2 
3 


= 


4.71 


47 


539.98 


x .0135157 


= 


7.29 


X 


I 


= 


4.86 


48 


529.12 


x .0142595 


= 


7.54 


X 


I 


= 


5.03 


49 


518.09 


x .0150611 


= 


7.80 


X 


t 


= 


5.20 


50 


506.90 


x .0159386 


= 


8.06 


X 


1 


— 


5.37 



A margin has already been made for expenses by 
considering the office premium $19.21, instead of 
$51.68, the net premium loaded thirty per cent. Re- 
ferring to Tables III. and IV. for the cost of assurance 
and the necessary reserves for each year, we will 
now lay before the reader a detailed statement of 
the account between the policy and the Company 
for the first sixteen years. The last item of debit 
in each year will be the amount of dividend due : — 

FIRST YEAR. 

Age, . . . Thirty -five. 

Policy No. Cr. 

By premium for the year, „....,. $49.21 

By interest at seven per cent 3.44 

$52.65 
Dr. 

To cost of assurance for the year $ 5 98 

To reserve at the end of the year , 34.78 

To dividend to balance 11.89 

$52.65 



31 



SECOND YEAR. 

Age, . . . Thirty-six. Cr. 

By reserve on hand at end of the first year $ 34.78 

By premium for the year 49.21 

By interest on premium and reserve 5.88 

S 89.87 
Dr. 

To cost of assurance for the year 8 5.87 

To reserve held at the end of the year 71.11 

To dividend to balance ~. 12.89 

8 89.87 

THIRD YEAR. 

Age, . . . Thirty-seven. Cr. 

By reserve at the end of the last year 8 71.11 

By premium for the year 49.21 

By interest 8.42 

8128.74 
Dr. 

To cost of assurance for the year 8 5.75 

To reserve at the end of the year 109.06 

To dividend to balance 13.93 

8128.74 

FOURTH YEAR. 

Age, . . . Thirty -eight. Cr. 

By reserve held at the end of the last year $109.06 

By premium for the year 49,21 

By interest 11.08 

8169.35 
Dr. 

To cost of assurance for the year $ 5.62 

To reserve at the end of the year 148.74 

To dividend to balance 14.99 

8169.35 

FIFTH YEAR. 

Age, . . . Thirty -nine. Cr. 

By reserve held at the end of the last year.. $148.74 

By premium for the year 49.21 

By interest 13.86 

$211.81 



32 

Dr. 

To cost of assurance for the year $ 5.46 

To reserve at the end of the year 190.23 

To dividend to balance.... 16.12 



$211 81 

SIXTH YEAE. 

Age, .' Forty. Or. 

By reserve held at the end of the last year , $190.23 

By premium for the year 49.21 

By interest 16.76 

$256.25 
Dr. 

To cost of assurance for the year $ 5.29 

To reserve at the end of the year , 233.64 

To dividend to balance 17.32 



$256.25 

SEVENTH TEAK. 

Age, .... Forty-one. Cr. 

By reserve held at the end of the last year . . . $233.64 

By premium for the year , .... 49.21 

By interest 19.80 

$302.65 
Dr. 

To cost of assurance for the year $ 5.10 

To reserve at the end of the year 279.08 

To dividend to balance 18.47 



02.65 

EIGHTH YEAR. 

Age, ... Forty-two. Cr. 

By reserve held at the end of the last year $279.08 

By premium for the year 49.21 

By interest 22.98 

$351.27 
Dr. 

To cost of assurance for the year $ 4.89 

To reserve at the end of the year 326.66 

To dividend to balance , 19.72 



$351.27 

NINTH YEAR. 

Age, . . . Forty-three. Cr. 

By reserve held at the end of the last year $326.66 

By premium for the year 49.21 

By interest 26.31 

$402.18 



33 

Dr. 

To cost of assurance for the year $ 4.67 

To reserve for the year 376.45 

To dividend to balance 21.06 

*402.18 

TENTH YEAB. 

Age, . . . Forty- four. Cr. 

By reserve held at the end of the last year $376.45 

By premium for the year 49.21 

By interest •., 29.79 

$455.45 
Dr. 

To cost of assurance for the year. B ... t ., ...= $ 4.46 

To reserve at the end of the year. = = .. 428 57 

To dividend to balance 22.42 



• 8455.45 

* ELEVENTH YEAR. 

Age, . . , . Forty-five. Cr. 

By reserve held at the end of the last year................. $428.57 

By interest..., „..o..„. . .==.=,. ===,==== 30.00 

8458.57 
Dr. 

To cost of assurance for the year = = = = = ,...= = . $ 4.57 

To reserve at the end of the year = ........ 438 86 

To dividend to balance ==.=.= , = = = . 15 14 



8458.57 

TWELFTH TEAB. 

Age, . p o Forty -six. Cr. 

By reserve held at the end of the last year . » 8438 86 

By interest = .,..., = = . = ..... 6 30.71 



$469.57 

Dr. 

To cost of assurance for the year ., =. .=... „...==, $ 4.71 

To reserve at the end of the year...... ...., 449.35 

To dividend to balance... . = ........... 15.51 



8469.57 



* The premiums being now paid in mil, we have on the credit side only 
the reserve from the end of the tenth year, improved at interest. 



31 



THIRTEENTH YEAR. 

Age, Forty-seven. Cr, 

By reserve held at the end of the last year $449.35 

By interest 31.47 



8480.82 
Dr. 

To cost of assurance for the year $ 4.86 

To reserve at the end of the year 460.02 

To dividend to balance 15.94 



$480. 82 



FOURTEENTH YEAR. 

Age, Forty -eight. Cr. 

By reserve held at the end of the last year , 8460.02 

By interest 32.20 



8492.22 
Dr. 

To cost of assurance for the year § 5.03 

To reserve at the end of the year..., 470.88 

To dividend to balance 16.31 



$492.22 

FIFTEENTH YEAR. 

Age, .... Forty-nine. Cr. 

By reserve held at the end of the last year $470.88 

By interest 32.96 

$503.84 

Dr. 

To cost of assurance for the year $ 5.20 

To reserve at the end of the year 481.91 

To dividend to balance ^ 16.73 

$503.84 

SIXTEENTH YEAR. 

Age, Fifty. Cr. 

By reserve held at the end of the last year $481.91 

By interest 33.73 

$515.64 
Dr. 

To cost of assurance for the year $ 5.37 

To reserve at the end of the year , 493.10 

To dividend to balance 17.17 

$515.64 



35 



These conditions are fully as favorable as have for 
any considerable length of time been experienced 
by any Company. It would be impossible, even if 
we knew the rate of interest that could be realized 
on investments, to predict the amount of future divi- 
dends, as the mortuary experience is very fluctuating 
and the expenses vary materially from year to year. 
Judging from past experience, it is probable that, 
taking the years together, the dividends in the 
six per cent, column of Table V. are fully as large 
as can be safely predicated for the future. 

In order to show the effect of the realization of a 
rate of interest in excess of that used in the compu- 
tation of premiums, w r e have given below a table in 
which a comparative statement is made between the 
dividends at seven per cent., as previously given, 
and five and six per cent., based upon the same 
assumptions in respect of mortality and loading. 







TABLE 


V. 






Four 


Five 


Six 


Seven 




per cent. 


per cent. 


per cent. 


per cent. 


1 


$10.42 


$10.91 


811.40 


£11.89 


2 


10.37 


11.21 


12 05 


12.89 


3 


10.32 


11.53 


12.73 


13.93 


4 


10.24 


11.82 


13.34 


14.99 


5 


10.17 


12.16 


14.14 


16.12 


6 


10.09 


12.48 


14.88 


17.32 


7 


9.98 


12.81 


15.62 


18.47 


8 


9.87 


13.15 


16.54 


19.72 


9 


9.78 


13.54 


17.30 


21.06 


10 


9.65 


13.91 


17.96 


22.42 


11 


2.28 


6.67 


10.94 


15.14 


12 


2.36 


6.74 


11.11 


15.51 


13 


2.43 


6.94 


11.43 


15,94 


14 


2.53 


7.11 


11.71 


16.31 


15 


2.60 


7.36 


12.02 


16.73 


16 


2.70 


7.54 


12.35 


17.17 



It is a noteworthy fact that the surplus in the four 
per cent, column arises wholly from the loading and 



36 



mortuary experience assumed. Below is a table in 
which, with the same assumptions for loading and 
expenses as in the preceding table, we give dividends 
computed at four, five, six and seven per cent., upon 
the supposition that the mortuary experience is pre- 
cisely that of the "rate" of mortality from which 
the premiums were calculated : — 

TABLE VI. 



Year. 


Four 


Five 


Six 


Seven 




per cent. 


per cent. 


per cent. 


per cent. 


1 


£7.44 


8 7.93 


$ 8 42 


| 8.90 


2 


7.44 


8.30 


9.14 


9.98 


3 


7.44 


8.66 


9.86 


11.16 


4 


7.44 


9.01 


10.53 


12.18 


5 


7.44 


9.44 


11.42 


13.40 


6 


7.44 


9.83 


12.23 


14.67 


7 


7.44 


10.26 


13.07 


15.92 


8 


7.44 


10.71 


14.10 


17.28 


9 


7.44 


11.21 


14.97 


18.73 


10 


7.44 


11.69 


15.94 


20.20 


11 


0.00 


4.29 


8 57 


12.86 


12 


0.00 


4.39 


8.78 


13.17 


13 


0.00 


4.49 


8.99 


13.48 


14 


0.00 


4.60 


9.20 


13.80 


15 


0.00 


4.71 


9.42 


14.13 


16 


0.00 


4.82 


9.64 


14.46 



In the four per cent, column the dividends arise 
wholly from the loading. In the five, six and seven 
per cent, columns it comes from both loading and 
interest. 

An analysis of the dividends of any year will 
show T the portion that has been contributed by the 
margin, the excess of interest, and the increased 
vitality of the assurants. By way of illustration, we 
will take the dividend of the second year, given in 
the seven per cent, column of Table V , the amount 
of which was $12.89:— 



37 



CONTRIBUTION FROM THE MARGIN. 

The office premium was $49.21 

The net premium was 42 06 

The margin was 7.15 

Improved at seven per cent, is the contribution 7.65 

CONTRIBUTION FROM INTEREST. 

The reserve was $34.78 

The net premium was 42.06 

The amount was 76.84 

The amount of interest received at seven per cent, was 5.38 

Amount required by our assumption at four per cent, was 3.07 

Contribution 2.31 

CONTRIBUTION FBOM VITALITY. 

Tabular cost of assurance was $ 8.80 

Actual cost was 5.87 

Contribution 2.93 

RECAPITULATION. 

Contribution from the margin $ 7.65 

Contribution from interest 2.31 

Contribution from increased vitality 2.93 

Whole amount of contribution ,.;.... 12.89 

In making up the interest account, it will be ob- 
served that we took the sum of the reserve and the 
net premium as the fund, as we had already im- 
proved the loading at the rate of interest actually 
received ; to have brought it into the general fund 
would have given us a twofold gain from its 
interest. 

An objection has been urged against the Contri- 
bution Plan on account of its complexity, and the 
amount of labor involved in making the necessary 
computations. This objection inures with equal 
force against all corporations that conduct an extend- 
ed business. Indeed, the amount of labor required 
is not as great as might be supposed, as, from the 
peculiar organization of an Assurance society, it 
admits of extended abridgments, of which ordinary 
business transactions are not susceptible. 



38 

It will be noticed that the dividends are adjusted 
to the actual mortuary experience of the Company. 
In order to ascertain what this experience is, it will 
be necessary to classify all members according to 
their ages, regardless of the time at which they were 
assured. This will enable the Company to assess the 
premiums paid by the individual members of any 
class to meet the losses that have occurred in that 
class between the periodic distributions. This im- 
plies a breadth of base which exists only in the larger 
Companies ; in smaller Companies the mortality will 
not be sufficiently uniform, particularly upon the 
extreme ages, to enable the Company to proceed in 
this manner. One of two alternatives is then left, 
viz. : to ascertain the per-centage of mortality on all 
ages taken together in comparison with the table ; or 
to take an average per cent, of the past mortuary 
experience of the Company upon any given age, 
and to increase or diminish this per-centage as the 
general mortality for the given year upon all ages 
is greater or less. 

Having ascertained the mortality upon the differ- 
ent ages, and the per-centum of assessment, we can 
proceed with another system of classification. All 
policies of the same sort — that is, policies issued dur- 
ing the same fiscal year, upon the same plan, and to 
persons of the same age — may be grouped together 
and treated as one large policy, representing the ag- 
gregate amount of all the smaller ones. The divi- 
dend on this large policy can be ascertained, and 
each of the smaller policies credited with its pro- 
portion. By this process the work of computation 
can be abridged, as we need have only as many ac- 
counts as we have different sorts of policies. 

The reader cannot fail to have observed that the 
dividends, with the exception of the four per cent, 
column in Table I V., uniformly increase till the end 



39 

of the tenth year ; that they decrease at the end of 
the eleventh year, and then constantly increase. 
This increase continues till the policy becomes a 
claim. The decrease at the end of the eleventh year 
is owing to the fact that, there being no payments 
after the tenth year, during that and the succeeding 
years no surplus arises from the margin or loading 
of the premium. The dividends succeeding the 
tenth arise, in Table V., from the excess of interest 
and from very favorable mortuary experience, and, 
in Table VI., from excess of interest alone. 

The facts developed in the tables show the in- 
equity of the uniform per-centage plan of distribut- 
ing surplus. The younger members would receive 
more than was their due, at the expense of the older 
ones. Practically, not only is this fact true, but the 
policies on one plan take part of the fund earned by 
policies on another, and the assured at one age the 
over-payments of those of another ; for the loading of 
the premiums on different plans is not always the 
same, and the mortality of different ages is often very 
far from holding the same relative correspondence to 
the table from which the premiums were computed. 

It is customary to hold at the end of the year, in 
addition to the reserve, a small fund to guard 
against adverse contingencies. This, in Companies 
making an annual distribution to all policies in 
force at the end of the year, is usually about four 
per cent. 

The system of distribution that we have given is 
very nearly equitable, but not exactly so. At the 
outset none but healthy lives are accepted, but, once 
taken, the Company is bound by the conditions of 
the contract to retain them. The physical condi- 
tion of the assured gradually deteriorates, not only 
from the inroads of disease, but from the fact that, 
as a rule, only the best risks withdraw. Thus the 



40 

Compaii} 7 , no matter how carefulty it makes its selec- 
tion at the outset, in time has a large number of im- 
paired risks on its hands. The mortality among 
these is greater than among those who have recently 
been assured. Grouping the old and new members 
together, and assessing all equally to pay the losses, 
would bear too heavily upon the new ones. This is 
a matter of secondary importance, but still it is 
worthy of remark. 

The influence of occupation upon the duration of 
life is a more noteworthy matter, which, while it 
has an important bearing upon the subject of assur- 
ance, is at present imperfectly understood. Noth- 
ing is more certain than that some employments 
tend to prolong life, while others shorten it. To a 
certain extent, this is a recognized element in the de- 
termination of the cost of assurance. Extra rates 
are charged to those who are engaged in hazardous 
employments, but no abatement is made to those 
who are surrounded by the most favorable sanitary 
conditions. It is diflicult, from the want of proper 
data, to determine with any degree of accuracy to 
what extent this should affect the cost of assurance. 
While Companies substantially agree in their pre- 
miums for healthy assurants, engaged in the ordinary 
vocations of life, their extra rates for hazardous risks 
vary materially. With increased diffusion of light, 
the practice of Companies will undoubtedly be 
modified. 

Life Assurance Societies are an outgrowth of 
modern civilization. Their aid is invoked to avert 
the financial calamity incident upon the failure of 
a productive life. By depositing annually a small 
sum, one can not only make provision for his declin- 
ing years, but, in the event of premature death, for 
those who are dependent upon him. By no other 
system can this be accomplished. 



41 

The vast accumulations held by these institutions 
are trust funds for the widow and the fatherless. 
The little deposits, coming back at a time when they 
are most needed, affording sustenance to childhood 
and support to old age, seem almost to be freighted 
with a double blessing. 

With so great an outreach into the future, com- 
mon prudence would seem to suggest that all the 
guards which human foresight could devise should 
be thrown around these institutions. The best 
scientific and financial talent should be called to the 
management of their affairs. Their security should 
be beyond question. Solvency first, and then equity, 
should be their watchwords. It is only by a high 
sense of the responsibility resting upon them, and by 
fidelity to the high trusts committed to them, that 
these institutions can secure the accomplishment of 
the benign mission which it is their purpose to 
fulfill. 



42 



TABLE I. 

Net Rates — "Actuaries' " Four Per Cent. Annuities and Single 
Premiums. 



Present Value of $1.00 per Annum, to 

be received at the commencement of 

every Year during Life. 


| Single Premium for%vhole Life Assurance 
for One Thousand Dollars. 


Age. 




Age. 


1 


Age. 


1 A S e - J 


10 


$20,454 


43 


$15,374 


10 


$213.32 


43 


■ $408.71 


11 


20.369 


44 


15.119 


11 


216.56 


44 


I 418.52 


12 


20.282 


45 


14.857 


12 


219.93 


45 


\ 428.57 


13 


20.191 


46 


14.590 


13 


223.44 


46 


1 438.86 


14 


20.096 


47 


14.317 


14 


227.08 


47 


449.35 


15 


19.998 


48 


14.039 


15 


230.86 


48 


460.02 


16 


19.896 


49 


13.757 


16 


234.78 


49 


470.88 


17 


19.790 


50 


13.470 


17 


238.84 


50 


48191 


18 


19.681 


51 


13.179 


18 


243.05 


51 


493.11 


19 


19.587 


52 


12.884 


19 


247.40 


52 


504.46 


20 


19.450 


53 


12.585 


20 


251.91 


53 


515.95 


21 


19.330 


54 


12.283 


21 


256.56 


54 


527.57 


22 


19.204 


55 


11.978 


22 


261.38 


55 


539.31 


23 


19.075 


56 


11.670 


23 


266.36 


56 


551.16 


24 


18.941 


57 


11.359 


24 


271.50 


57 


563.10 


25 


18.803 


58 


11.046 


25 


276.82 


58 


575.14 


26 


18.660 


59 


10.731 


26 


282.31 


59 


587.26 


27 


18.512 


60 


10.415 


27 


287.99 


60 


599.43 


28 


18.360 


61 


10.098 


28 


293.86 


61 


611.63 


29 


18.202 


62 


9.780 


29 


299.91 


62 


623.83 


30 


18.040 


63 


9.464 


30 


306.17 


63 


636.00 


31 


17.872 


64 


9:149 


31 


312.62 


64 


648.12 


32 


17.698 


65 


8.835 


32 


319.29 


65 


660.17 


33 


17.520 


66 


8.525 


33 


326.17 


66 


672.13 


34 


17.335 


67 


8.217 


34 


333.27 


67 


683.96 


35 


17.144 


68 


7.913 


35 


340.60 


68 


695.66 


36 


16.948 


69 


7.613 


36 


348.17 


69 


707.19 


37 


16.744 


70 


7.317 


37 


355.99 


70 


718.57 


38 


16.534 


71 


7.026 


38 


364.07 


71 


729.76 


39 


16.317 


72 


6.740 | 


39 


372.42 


72 


740.76 


40 


16.093 


73 


6.459 


40 


381.04 


73 


751.57 


41 


15.861 


74 


6.184 


41 


389.96 


74 


762.15 


42 


15.621 


75 


5.915 


42 


399.18 


75 


772.51 



43 



TABLE II. 

Net Rates — "Actuaries' " Four Per Cent. Premiums for a Whole 
Life Assurance of One Thousand Dollars. 





By Annual Payme 


NTS. 


i Bl 


' Ten Anxva 


l Payments. 


Age. 




Age. 




Age. 


A g e - j 


10 


10.43 


43 


26.58 


10 


26.02 


43 


51.08 


11 


10.63 


44 


27.68 


11 


26.42 


44 


52.44 


12 


10.84 


45 


28.84 


12 


26.83 


45 


53.86 


1 13 


11.07 


46 


30.08 


13 


27.27 


46 


55.33 


14 


11.30 


47 


31.38 


14 


27.72 


47 


56.85 


15 


11.54 


48 


32.77 


15 


28.19 


48 


58.43 


16 


11.80 


49 


34.23 


16 


28.68 


49 


60.05 


.17 


12.07 


50 


35.78 


17 


29.18 


50 


6174 


18 


12.35 


51 


37.42 


18 


. 29.71 


51 


63.49 


19 


12.64 


52 


39.15 


19 


30.25 


52 


65.30 


20 


12.95 


53 


40.97 


20 


30.81 


53 


67.17 


21 


13 27 


54 


42.95 


21 


31.40 


54 


69.12 


22 


13.61 


55 


45.03 


22 


32 00 


55 


71.14 


23 


13.96 


56 


47.23 


23 


32.63 


56 


73.25 


24 


14.33 


57 


49.57 


24 


33 27 


57 


75.44 


25 


14.72 


58 


52.07 


25 


33.94 


58 


77.74 


26 


15.13 


59 


54.72 


26 


34.64 


59 


80.15 


27 


15.58 


60 


57.56 


27 


35.35 


60 


82 68 


28 


16.00 


61 


60.57 


28 


36.09 


61 


85.34 


29 


16.48 


62 


63.78 


29 


36.86 


62 


88.13 


30 


1697 


63 


67.20 


30 


37.66 


63 


91.07 


31 


17.49 


64 


70.84 


31 


38.48 


64 


94.16 


32 


18.04 


65 


74.72 


32 


39.33 


65 


97.43 


33 


18.62 


66 


78.85 


33 


40.21 


66 


100.88 


34 


19.23 


67 


83.24 


34 


41.12 


67 


104.53 


35 


19.89 


68 


87.91 


35 


42.06 


68 


108.39 


36 


20.54 


69 


92.89 


36 


43.04 


69 


112.48 


37 


21.26 


70 


98.20 


37 


44.05 


70 


116.85 


38 


22.02 


71 


103.87 


38 


45.10 


71 


121.50 


39 


22.82 


72 


109.91 


39 


46.20 


72 


126.48 


40 


23.68 


73 


116.36 


40 


47.34 


73 


131.81 


i 41 


24.59 


74 


123.25 


41 


48.53 


74 


137.53 


1 42 


25.55 


75 


130.48 


42 


49.77 


75 


143 68 



u 



TABLE III. 

Net Rates "Actuaries Four Per Cent. 
ENDORSEiMENT ASSURANCE, 

Annual Premiums for the whole term for each One Thousand 
Dollars, assured and endorsed, payable at death or at the age of 



Age. 


35 


40 


45 


BO 


65 

$13.75 


60 


65 


70 


10 


$27.58 


$21.96 


$18 19 


$15.58 


$12.47 


$11.60 


$11.03 


11 


29.07 


22.95 


18.90 


16.10 


14.15 


12.79 


11.87 


1127 


12 


30.70 


24.03 


19.65 


16.67 


14.58 


13.13 


12.15 


11.52 


13 


32 49 


25.19 


20.46 


17.24 


15.02 


13.49 


12.45 


11.78 


14 


34.47 


26.46 


21.32 


17.87 


15.50 


13.87 


12.76 


12.05 


15 


36.66 


27.83 


22.26 


18.55 


16 01 


14.27 


13.10 


12 34 


16 


39.10 


29.33 


23.16 


19.26 


1^54 


14.69 


13.44 


12.64 


17 


41.82 


30.98 


24.35 


20.03 


17.12 


15.14 


13.81 


12 96 


18 


44.88 


32.78 


25.53 


20.85 


17.72 


15.61 


14.20 


13.30 


19 


48.33 


34.77 


26.81 


21.73 


18.37 


1611 


14.61 


13 65 


20 


52.27 


36.97 


28.19 


22.68 


19.C6 


16.64 


15.04 


14.02 


21 


56.79 


39.42 


29 71 


23.70 


19.80 


17.20 


15.49 


14.41 


22 


62.02 


42.15 


31.36 


24.80 


20.58 


17.80 


15.97 


14.81 


23 


68.14 


45.21 


33.18 


25.99 


21.42 


18.43 


16.48 


15^25 


24 


75.41 


48.68 


35.18 


27.28 


22.32 


19.10 


17.01 


15.70 


25 


84.15 


52.62 


37 38 


28.69 


23.29 


19.82 


17.58 


16.17 


26 


94.*7 


57.15 


39.84 


30.21 


24.33 


20 58 


18.18 


16.68 


27 


108.29 


62.38 


42.58 


31.88 


25.45 


21.40 


18.81 


1721 


28 


125.59 


68.52 


45 66 


33.71 


26.67 


22.27 


19.49 


17.77 


23 


148.71 


75.79 


49.13 


35.72 


27.98 


23.21 


20.21 


18.36 


30 


181.11 


84.53 


53.08 


37.95 


29.40 


24.21 


20.&7 


18.99 


31 


229.80 


95.26 


57.62 


40.42 


30.96 


25.29 


21.78 


19.65 


32 




108.69 


62.86 


43.71 


32.65 


26.44 


22 65 


20.36 


33 




125.99 


69.00 


46.26 


34.51 


27.70 


23.58 


21.10 


34 




149.10 


76.28 


49.75 


36.55 


29.05 


24.57 


21.89 


35 




181.60 


85.03 


53.72 


38.80 


30.52 


25.63 


22.74 


36 




230.16 


95.76 


58.27 


41*30 


32.12 


26.78 


23.64 


37 






109.19 


63.53 


44.09 


33.87 


2^.01 


24.60 


38 






126.49 


69.69 


47.22 


35.78 


29.34 


25.63 


39 






149.59 


76.99 


50.75 


37.89 


30.78 


26.73 


40 






181.98 


85.76 


54.77 


40.21 


32.35 


27.92 


41 






230.62 


96.52 


59.38 


42.80 


34.05 


29.19 


42 








109.98 


6471 


45.68 


35.92 


39.57 


43 








127.33 


70.95 


48 92 


37.96 


32 C6 


44 








150.49 


78.34 


52.56 


40.21 


33 67 


45 








182.92 


87.21 


56.70 


42.68 


35.42 


46 








231.60 


98.05 


61.43 


45.41 


37.31 


47 










111.61 


66.87 


48.44 


39 35 


48 










129.02 


73.22 


51.81 


41.58 


49 










152.21 


80.70 


5559 


44.C0 


50 










184.66 


89.66 


59.86 


46.65 


51 










233.29 


100.59 


64.72 


49.57 


52 












114.19 


70.29 


52.78 


53 












131.66 


76.76 


56.35 


54 












154.89 


84.S7 


60.33 


55 












187.34 


93.45 


64.80 


56 












235.90 


104.48 


69.87 


57 














108.20 


75.(5 


58 














135.75 


82.34 


59 














159.04 


90.17 


60 














191.49 


99.47 


61 














236.97 


110.72 


62 
















124.64 


63 
















142.35 


64 
















165.73 


65 
















198.19 


66 
















246.49 



45 



Table I. 
COMPOUND INTEREST, 

Showing the amount of $1, improved at Compound Interest, for any 
number of years not exceeding 100. 



Years. 4 per Ct. 4^p , rCt.j 5perCt. 6perCt. 7perCt. 8perCt. 



9 
10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 

31 
32 
33 
34 



37 



1.040000 
1.081600 
1.124864 
1.169859 

1 216653 

1.265319 
1.315932 
1.368569 
1.423312 
1.480244 

1.539454 
1.601032 
1.665074 
1.731676 

1.800944 

1.872981 
1.947901 
2.025817 
2.106849 
2.191123 

2.278768 

2 369919 
2.464716 
2.563304 
2.665836 

2.772470 
2.883369 
2.998703 
3.118651 
3.243398 

3.373133 
3.508059 
3.648381 
3.794316 
3.946089 



1.045000 | L050C00 
1.092025 1.102500 



1.141166 
1.192519 
1.246182 

1.302260 
1.360862 
1.422101 

1.486095 
1.552969 



1.772196 
1.851945 
1.935282 

2.022370 
2.113377 
2 208479 
2.307860 
2.414714 

2.520241 
2.633652 
2.752166 
2.876014 
3.005434 

3.140679 
3.282010 
3.429700 
3.584036 
3.745318 

3.913857 
4.089981 
4.274030 
4.466362 
4.667348 



1.157625 
1.215506 

1.276282 

1.340096 
1.407100 
1.477455 
1.551328 
1.628895 



1.622853 1.710339 
1.695881 1.795856 



4.103933 4.877378 
4.268090 I 5.096860 



4.438813 
4.616366 
4.801021 

4.993061 
5.192784 
5.400495 
5.616515 
5.841176 

6.074823 
6.317816 
6.570528 
6.833349 
7.106683 



5.326219 
5.565899 
5.816365 

6.078101 
6.351615 
6,637438 
6.936123 
7.248248 

7.574420 
7.915268 
8.271456 
8.643671 
9.032636 



1.885649 
1.979932 
2 078928 

2.182875 2.540352 
2.292018 2.692773 



1 060000 
1.123600 
1.191016 
1.262477 
1.338226 

1.418519 
1.503630 
1.593848 
1.689479 
1.790848 

1.898299 
2.012196 

2.132928 
2.260904 
2.396558 



2.406619 
2.526950 
2.653298 

2.785963 
2.925261 
3.071524 
3.225100 
3.386355 

3.555673 
3.733456 
3.920129 
4.116136 
4.321942 

4.538039 
4 764941 
5.003189 
5.253348 
5.516015 

5.791816 
6.081407 
6,385477 
6.704751 
7.039989 



7.761588 
8.149667 
8.5571iO 

8.9:50^8 

9.434258 
9.905971 
10.401270 
10.921333 
11.467400 



2.854339 
3.025600 
3.207135 

3.399564 
3.603537 
3.819750 
4.048935 
4.291871 

4.549383 
4.822346 
5.111687 
5.4L8388 
5.743491 

6.08*101 
6.453387 
6.840590 
7.251025 
7.686087 

8.147252 
8.636087 
9.154252 
9.703507 
10.285718 

10.902861 
11.557033 
12.250455 
12 985482 
13.764611 

14.590487 
15.465917 
16.3-3872 
17.377604 
18.420154 



1.070000 
1.144900 
1.225043 
1.310796 
1.402552 

1.500730 
1.605781 
1.718186 
1.838459 
1.967151 

2.104862 
2.252192 
2.409845 
2.578534 
2.759032 

2.952164 

3.15881c 
3.379932 
3.616528 
3.869684 

4.140562 
4.430402 
4.740530 
5.072367 
5.427433 

5.807353 
6.213868 
6.64883S 
7.114257 
7.612255 

8.145113 
8.715271 
9.325340 
9978114 
10.676581 

11.423942 
12.223618 
13.079271 
13.994820 
14.974458 



1.080000 
1.166400 
1.259712 
1.360489 
1.469328 

1.586874 
1.713824 
1.850930 
1.999005 
2.158925 

2.331639 
2.518170 
2.719624 
2.937194 
3.172169 

3.425943 
3.700018 
3.996020 
4.315701 
4.660957 

5.033834 
5.436540 
5.871464 
6.341181 

6.848475 

7.396353 

7.988061 
8.627106 
9.317275 
10.062657 

10.867669 
11.737083 
12.676050 
L3.690134 
14.785344 

15.968172 
17.246626 
18.625276 
20.115298 
21.724522 



16.022670 
17.144257 
18.344355 
19.628460 
21.002452 

22.472623 
24.045707 

25.728907 
27.529930 
29.457025 



23.462483 
25.339482 
:7. 366640 
29.555972 
31.920449 

34.474085 
37.232012 
40.210573 
43.427419 
46.901613 



46 



Table II. 
COMPOUND INTEREST. 

The amount of%\ per annum in any number of Years. 



Years. 


4 per Cent. 


5 per Cent. 


6 per Cent. 


| 7 per Cent. 


8 per Cent- 


1 


1 .000000 


1.000000 


1.000000 


1.000000 ' 


1.000000 


2 


2.04C00O 


2.050000 


2.060000 


2 070000 


2.080000 


3 


3J21600 


3.152500 


3.183600 


3.214900 


3.246400 


4 


4.246461 


4.310125 


4.374616 


4.439943 


4.506112 


5 


5.416323 


0.525631 


5.637093 


5.750739 


5.866601 


6 


6.632975 


6.801913 


6.975319 


7.153291 


7.335929 


7 


7.898294 


8.14-2008 


8.393838 


8 654021 


8.922803 


8 


9.214226 


9.549109 


9.897468 


10.259803 


10.636628 


9 


10.582795 


11.026564 


11.491316 


11.977989 


12.487558 


10 


12.006107 


12.577893 


13.180795 


13.816448 


14.486562 


11 


13.486351 


14.206787 


14.971613 


15.783599 


16 645487 


12 


15.025805 


15.917127 


16 869941 


17.888451 


18.977126 


13 


16.626838 


17.712983 


18.882138 


20 140643 


21.495297 


14 


18.291911 


19.598632 


21.015066 


22.550488 


24.214920 


15 


20.023588 


21.578561 


23 275970 


25.129022 


27.152114 


16 


21.824531 


23.657492 


25.672528 


27.888054 


30.324283 


17 


23.697512 


25.840366 


28.212880 


30.840217 


33.750226 


18 


25.645413 


28.132385 


30.905653 


33.999033 


37.450244 


19 


27.671229 


39.539004 


33.759992 


37.378965 


41.446263 


20 


29.778079 


33.065954 


36.785591 


40.995492 


45.761964 


21 


31.969202 


35.719252 


39.992727 


44.865177 


50.422921 


22 


34.247970 


38.505214 


43.392290 


49.005739 


55.456755 


23 


36.617889 


41.430475 


46.995828 


53.436141 


60.893296 


24 


39.082604 


44.501999 


50.815577 


58.176671 


66.764759 


25 


41.645908 


47.727099 


54.864512 


63.249038 


73.105940 


26 


44.311745 


51.113454 


59.156383 


68.676470 


79.954415 


27 


47.084214 


54.669126 


63.705766 


74.483823 


87.35076S 


28 


49.967583 


58.402583 


68.528112 


80.697691 


95 338830 


29 


52.966286 


62.322712 


73.639798 


87.346529 


103.965936 


\ 30 


56.084938 


66.438848 


79.058186 


94.460786 


113.283211 


31 


59.328335 


70.760790 


84 801677 


102.073041 


123.345868 


32 


62.701469 


75.298829 


90.889778 


110.218154 ■ 


134.213537 


33 


66.209527 


80.063771 


97-343165 


118.933425 


145.950620 


34 


69.857909 


85.066959 


104.183755 


128.258765 


158.626670 


35 


73.652225 


90.320307 


111.434780 


138.236578 


172,316804 


36 


77.593314 


95.836323 


119.120867 


148.913460 


187.102148 


37 


81.702246 


101.628139 


127.268119 


160.337402 


203.070320 


38 


85.970336 


107.709546 


135.904206 


172.561020 


220 315945 


39 


90.409150 


114.095023 


145.058458 


185.640292 


238.941221 


40 


95.025516 


120.799774 


154.761966 


199.635112 


259.056519 


41 


99.826536 


127.839763 


165.047684 


214.609570 


280.781040 


42 


104.819598 


135 231751 


175.950545 


230.632240 


304.243523 


43 


110.012382 


142.993339 


187.507577 


247.776496 


329.583005 


44 


115.412877 


151.143006 


199.758032 


266 120851 


356.949646 


45 


121.029392 


159.700156 


212.743514 


285.749311 


386.505617 


46 


126.870568 


168,685164 


226.508125 


306.751763 


418.426067 


47 


132.945390 


178.119422 


241.098612 


329.224386 


452.900152 


48 


139.263206 


188.025393 


255.564529 


353 270093 


490 132164 


49 


145.833734 


199.426663 


272.958401 


378.999000 


530.342737 


50 


152.667084 


209.347996 


290.335905 


406.528929 


573.770156 



47 



Table III. 

THE PRESENT VALUE OF ONE DOLLAR PER ANNUM, 

For any number of years (o 50, at 4, 5, 6, 7 and 8 per 
cent, interest. 



Years. 


4 per Cent. 


5 per Cent. 


6 per Cent. 


7 per Cent. 


8 per Cent. 


1 


.961538 


.952381 


.943396 


.934579 


.925926 


2 


1.886095 


1.859410 


1.833393 


1.808018 


1.783265 


3 


2.775091 


2.723248 


2.673012 


2.624316 


2.577097 


4 


3.629895 


3.545951 


3.465106 


3.387211 


3.312127 


5 


4.451822 


4.329477 


4.212364 


4.100197 


3.992710 


6 


5.242137 


5.075692 


4.917324 


4.766540 


4.622880 


7 


6.002055 


5.786373 


5.582381 


6.389289 


5.206370 


8 


6.732745 


6.463213 


6.209794 


5.971299 


5.746639 


9 


7.435332 


7.107822 


6.801692 


6.515232 


6.246888 


! 10 


8.110896 


7.721735 


7.360087 


7.023582 


6.710081 


11 


8.760477 


8-306414 


7.886875 


7.498674 


7.138964 


12 


9.385074 


8.863252 


8.3S3844 


7.942686 


7.536078 


13 


9.985648 


9.393573 


8.852683 


8.357651 


7.903776 


14 


10.563123 


9.898641 


9.294984 


8.745468 


8.244237 


15 


11118387 


10.379658 


9.712249 


9.107914 


8.559479 


16 


11.652296 


10.837770 


10.105895 


9*446649 


8.851369 


17 


12.165669 


11.274066 


10.477260 


9'763223 


9.121638 


18 


12.659297 


11.689587 


10.827603 


10'059C87 


9.371887 


19 


13.133939 


12.085321 


11.158116 


10-335595 


9.603599 


20 


13.590326 


12.462210 


11.469921 


10-594014 


9.818147 


21 


14.029160 


12.821153 


11.764077 


10.835527 


10.016803 


22 


14.451115 


13.163003 


12 041582 


11.061241 


10.200744 


23 


14.856842 


13.488574 


12.303379 


11.272187 


10.371059 


24 


15.246963 


13.798642 


12.550358 


11.469334 


10.528758 


25 


15.622080 


14.093945 


12.783356 


11.653583 


10.674776 


26 


15.982769 


14.375185 


13.003166 


11.825779 


10.809978 


27 


16.329586 


14.643034 


13.210534 


11.986709 


10;935165 


28 


16.663063 


14.898127 


13.406164 


12.137111 


11.051078 


29 


16.983715 


15.141074 


13.590721 


12.277674 


11.158406 


30 


17.292033 


15.372451 


13.764831 


12.409041 


11.257783 


31 


17.588494 


15.592811 


13.929086 


12.531814 


11.349799 


32 


17.873552 


15.802677 


14.084043 


12.646555 


11.434999 


33 


18.147646 


16.002549 


14.230230 


12.753790 


11.513888 


34 


18.411198 


16.192904 


14.368141 


12.854009 


11.516934 


35 


18.664613 


16.374194 


14.498246 


12.947672 


11.654568 


36 


18.908282 


16.546852 


14.620987 


13-035208 


11.717193 


37 


19.142579 


16.711287 


14.736780 


13.117017 


11.775179 


38 


19.367864 


16.867893 


14.846019 


13.193473 


11.828869 


39 


19.584485 


17.017041 


14.949075 


13.264928 


11.878582 


40 


19.792774 


17.159086 


15.046297 


13.331709 


11.924613 


41 


19.993052 


17.294368 


15.138016 


13.394120 


11.967235 


42 


20.185627 


17.423208 


15.224543 


13.452449 


12.006699 


43 


20.370795 


17.545912 


15.306173 


13.506962 


12.043240 


44 


20.548841 


17.662773 


15.383182 


13.557908 


12.077074 


45 


20.720040 


17.774070 


15.455832 


13.605522 


12.108402 


46 


20.884654 


17.8S0087 


15.524370 


13.650020 


12.137409 


47 


21.042936 


17.981016 


15.589028 


13.691608 


12.164267 


48 


21.195131 


18.077158 


15.650027 


13.730474 


12.189136 


49 


21.341472 


18.168722 


15.707572 


13.766799 


12 212163 


50 


21.482185 


18.255925 


15.761861 


13.800746 


12.233485 



48 



Table IV. 

THE PRESENT VALUE OF ONE DOLLAR, 

Due at the end of any number of years to 100, at 4, 5, 6, 7 and S 
per cent, interest. 



Years 


4 per Cent. 


5 per Cent. 


6 per Cent. 


7 per Cent. 


8 per Cent. 


1 


.96153846 


.95238095 


.94339623 


.93459744 


.92592593 


2 


.9245^621 


.90702948 


.88999644 


.87343873 


.85733882 


3 


.88899636 


.86383760 


.83961928 


.81629788 


.79383224 


4 


.85480419 


.82270247 


.79209366 


.76289521 


.73502985 


5 


.82192711 


.78352616 


.74725817 


.71298618 


.68058320 


6 


.79031453 


.74621540 


.70496054 


.66634222 


.63016963 


7 


.75991781 


.71068133 


.66505711 


.62274974 


.58349040 


8 


.73069020 


.676*3936 


.62741237 


.58200910 


.54026888 


9 


.70258674 


.64460892 


.59189846 


.54393374 


.50024897 


10 


.67556417 


.61391325 


.55839478 


.50834929 


.46319349 


11 


,64958093 


.58467929 


.52678753 


.47509280 


.42888286 


12 


.62459705 


.55683742 


.49696936 


.44401196 


.39711376 


13 


.60057409 


.53032135 


.46883902 


.41496445 


.36769792 


14 


.57747508 


.50506795 


.44230096 


.38781724 


.34046104 


15 


.55526450 


.48101710 


.41726506 


.36244602 


.31524171 


16 


.53390818 


.45811152 


.39364628 


.33873460 


.29189047 


17 


.51337325 


.43629669 


.37136442 


.31657439 


.27026895 


18 


.49362812 


.41552065 


.35034379 


.29586392 


.25024903 


19 


.47464242 


.39573396 


.33051301 


.27650833 


.23171206 


20 


.45638695 


.37688948 


.31180473 


.25841900 


.21454821 


21 


.43883360 


.35894236 


.29415540 


.24151309 


.19865575 


22 


.42195539 


.34184987 


.27750510 


.22571317 


.18394051 


23 


.40572633 


.32557131 


.26179726 


.21094688 


.17031528 


24 


.39012147 


.31006791 


.24697855 


.19714662 


.15769934 


25 


.37511680 


.29530277 


.23299863 


.18424918 


.14601790 


26 


.36068923 


.28124073 


.21981003 


.17219549 


.13520176 


27 


.34681657 


.26784832 


.20736795 


.16093037 


.12518682 


28 


.33347747 


.25509364 


.19563014 


.15040221 


.11591372 


29 


.32065141 


.24294632 


.18455674 


.14056282 


.10732752 


30 


.30831867 


.23137745 


.17411013 


.13136712 


.09937733 


31 


.29646026 


.22035947 


.16425484 


.12277301 


.09201605 


32 


.28505794 


.209SC617 


.15495740 


.11474113 


.08520005 


33 


.27409417 


.19987254 


.14618622 


.10723470 


.07888893 


34 


.26355209 


.19035480 


,13791153 


.10021934 


.07304531 


35 


.25341547 


.18129029 


.13010522 


.09366294 


.06763454 


36 


.24366872 


.17265741 


.12274077 


.08753546 


.06262458 


37 


.23429685 


.16443563 


.11579318 


.08180884 


.05798572 


38 


.22528543 


.156(30536 


.10923885 


.07645686 


.05369048 


39 


.21662u61 


.14914797 


.10305552 


.07145501 


.04971341 


40 


20828904 


.14204568 


.09722219 


.06678038 


.04603093 


41 


.20027792 


.13528160 


.09171905 


.06241157 


.04262123 


42 


.19257493 


.12883962 


.08652740 


.05832857 


.03946411 


43 


.18516820 


.12270440 


.08162962 


.05451268 


.03654084 


44 


.17804635 


.11686133 


.07700903 


.05094643 


.03383411 


45 


.17119841 


.11129651 


.07265007 


.04761349 


.03132788 


46 


.16461386 


.10599668 


.06853781 


.04449859 


.02900730 


47 


.15828256 


10094921 


.06465831 


.04158747 


.02685861 


48 


.15219476 


.09614211 


.06099840 


.03886679 


.02486908 


49 


.14634112 


.09156391 


.05754566 


.03632410 


.02302693 


50 


.14071262 


.03720373 


.05428836 


.03394776 


.02132123 



